The conservation of momentum can be useful in designing control laws for underactuated mechanical systems. However, the momentum is conserved only for unactuated variables with symmetry. If a symmetry-breaking force is applied to a system, the momentum is not conserved any longer in general. The main objective of this paper is to show that there exist forces linear in velocity such as the damping force that break the symmetry but induce a new conserved quantity in place of the original momentum map. This paper formalizes that such a new conserved quantity can be constructed by combining the time integral of a general damping force and the original momentum map associated with the symmetry. Especially, we show that the new conserved quantity can exist for multiple variables with symmetry. From the perspective of stability theories, the major implication of the new conserved quantity is that the corresponding variables possess the self recovery phenomenon, i.e. they will be globally attractive to the initial condition of the variables. What is fundamental in the damping-induced self recovery is not the positivity of the damping coefficient but certain properties of the time integral of the damping force. Self recovery effect and theoretical findings are demonstrated by simulation results using a two-link manipulator and a planar pendulum. The results in this paper will be useful in designing and controlling mechanical systems with underactuation.

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